Spectral Universality of Real Chiral Random Matrix Ensembles

TL;DR

This paper demonstrates the universality of microscopic eigenvalue correlations in real chiral random matrix ensembles by relating them to complex ensembles and analyzing skew-orthogonal polynomials.

Contribution

It establishes a novel proof of universality for real chiral ensembles using asymptotic properties of skew-orthogonal polynomials, linking to known complex ensemble results.

Findings

Microscopic eigenvalue correlations are universal across different ensembles.

A new asymptotic property of skew-orthogonal polynomials is proven.

Numerical construction confirms analytical results.

Abstract

We investigate the universality of microscopic eigenvalue correlations for Random Matrix Theories with the global symmetries of the QCD partition function. In this article we analyze the case of real valued chiral Random Matrix Theories ($\beta =1$) by relating the kernel of the correlations functions for $\beta =1$ to the kernel of chiral Random Matrix Theories with complex matrix elements ($\beta = 2$), which is already known to be universal. Our proof is based on a novel asymptotic property of the skew-orthogonal polynomials: an integral over the corresponding wavefunctions oscillates about half its asymptotic value in the region of the bulk of the zeros. This result solves the puzzle that microscopic universality persists in spite of contributions to the microscopic correlators from the region near the largest zero of the skew-orthogonal polynomials. Our analytical results are illustrated by the numerical construction of the skew-orthogonal polynomials for an $x^4$ probability potential.